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Your data matches 558 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000245
(load all 162 compositions to match this statistic)

Mapping

St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 = 2 - 2
[1,2] => 1 = 3 - 2
[2,1] => 0 = 2 - 2
[1,2,3] => 2 = 4 - 2
[1,3,2] => 1 = 3 - 2
[2,1,3] => 1 = 3 - 2
[2,3,1] => 1 = 3 - 2
[3,1,2] => 1 = 3 - 2
[1,2,3,4] => 3 = 5 - 2
[1,3,2,4] => 2 = 4 - 2
[1,4,2,3] => 2 = 4 - 2
[2,1,3,4] => 2 = 4 - 2
[2,3,1,4] => 2 = 4 - 2
[2,4,1,3] => 2 = 4 - 2
[3,1,2,4] => 2 = 4 - 2
[3,4,1,2] => 2 = 4 - 2
[4,1,2,3] => 2 = 4 - 2
[1,2,3,4,5] => 4 = 6 - 2
[2,1,3,4,5] => 3 = 5 - 2
[3,1,2,4,5] => 3 = 5 - 2
[4,1,2,3,5] => 3 = 5 - 2
[5,1,2,3,4] => 3 = 5 - 2
[1,2,3,4,5,6] => 5 = 7 - 2

Description

The number of ascents of a permutation.

Matching statistic: St000672
(load all 252 compositions to match this statistic)

Mapping

St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 = 2 - 2
[1,2] => 1 = 3 - 2
[2,1] => 0 = 2 - 2
[1,2,3] => 2 = 4 - 2
[1,3,2] => 1 = 3 - 2
[2,1,3] => 1 = 3 - 2
[2,3,1] => 1 = 3 - 2
[3,1,2] => 1 = 3 - 2
[1,2,3,4] => 3 = 5 - 2
[1,3,2,4] => 2 = 4 - 2
[1,4,2,3] => 2 = 4 - 2
[2,1,3,4] => 2 = 4 - 2
[2,3,1,4] => 2 = 4 - 2
[2,4,1,3] => 2 = 4 - 2
[3,1,2,4] => 2 = 4 - 2
[3,4,1,2] => 2 = 4 - 2
[4,1,2,3] => 2 = 4 - 2
[1,2,3,4,5] => 4 = 6 - 2
[2,1,3,4,5] => 3 = 5 - 2
[3,1,2,4,5] => 3 = 5 - 2
[4,1,2,3,5] => 3 = 5 - 2
[5,1,2,3,4] => 3 = 5 - 2
[1,2,3,4,5,6] => 5 = 7 - 2

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Matching statistic: St000010
(load all 4 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => 1 = 2 - 1
[1,2] => [1,1]
 => 2 = 3 - 1
[2,1] => [2]
 => 1 = 2 - 1
[1,2,3] => [1,1,1]
 => 3 = 4 - 1
[1,3,2] => [2,1]
 => 2 = 3 - 1
[2,1,3] => [2,1]
 => 2 = 3 - 1
[2,3,1] => [2,1]
 => 2 = 3 - 1
[3,1,2] => [2,1]
 => 2 = 3 - 1
[1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[1,3,2,4] => [2,1,1]
 => 3 = 4 - 1
[1,4,2,3] => [2,1,1]
 => 3 = 4 - 1
[2,1,3,4] => [2,1,1]
 => 3 = 4 - 1
[2,3,1,4] => [2,1,1]
 => 3 = 4 - 1
[2,4,1,3] => [2,1,1]
 => 3 = 4 - 1
[3,1,2,4] => [2,1,1]
 => 3 = 4 - 1
[3,4,1,2] => [2,1,1]
 => 3 = 4 - 1
[4,1,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[2,1,3,4,5] => [2,1,1,1]
 => 4 = 5 - 1
[3,1,2,4,5] => [2,1,1,1]
 => 4 = 5 - 1
[4,1,2,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[5,1,2,3,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 6 = 7 - 1

Description

The length of the partition.

Matching statistic: St000062
(load all 23 compositions to match this statistic)

Mapping

Mp00329: Permutations —Tanimoto⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => 2 = 3 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 3 = 4 - 1
[1,3,2] => [2,1,3] => 2 = 3 - 1
[2,1,3] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [3,1,2] => 2 = 3 - 1
[3,1,2] => [2,3,1] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,3,2,4] => [1,2,4,3] => 3 = 4 - 1
[1,4,2,3] => [2,1,3,4] => 3 = 4 - 1
[2,1,3,4] => [1,3,2,4] => 3 = 4 - 1
[2,3,1,4] => [1,3,4,2] => 3 = 4 - 1
[2,4,1,3] => [3,1,2,4] => 3 = 4 - 1
[3,1,2,4] => [1,4,2,3] => 3 = 4 - 1
[3,4,1,2] => [4,1,2,3] => 3 = 4 - 1
[4,1,2,3] => [2,3,4,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[2,1,3,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[3,1,2,4,5] => [1,4,2,3,5] => 4 = 5 - 1
[4,1,2,3,5] => [1,5,2,3,4] => 4 = 5 - 1
[5,1,2,3,4] => [2,3,4,5,1] => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 6 = 7 - 1

Description

The length of the longest increasing subsequence of the permutation.

Matching statistic: St000213
(load all 81 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => 2 = 3 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 3 = 4 - 1
[1,3,2] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [2,1,3] => 2 = 3 - 1
[2,3,1] => [3,2,1] => 2 = 3 - 1
[3,1,2] => [2,3,1] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[2,3,1,4] => [3,2,1,4] => 3 = 4 - 1
[2,4,1,3] => [3,2,4,1] => 3 = 4 - 1
[3,1,2,4] => [2,3,1,4] => 3 = 4 - 1
[3,4,1,2] => [2,4,3,1] => 3 = 4 - 1
[4,1,2,3] => [2,3,4,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[2,1,3,4,5] => [2,1,3,4,5] => 4 = 5 - 1
[3,1,2,4,5] => [2,3,1,4,5] => 4 = 5 - 1
[4,1,2,3,5] => [2,3,4,1,5] => 4 = 5 - 1
[5,1,2,3,4] => [2,3,4,5,1] => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 6 = 7 - 1

Description

The number of weak exceedances (also weak excedences) of a permutation. This is defined as

$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of

$\sigma$ .

Matching statistic: St000308
(load all 12 compositions to match this statistic)

Mapping

Mp00329: Permutations —Tanimoto⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => 2 = 3 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 3 = 4 - 1
[1,3,2] => [2,1,3] => 2 = 3 - 1
[2,1,3] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [3,1,2] => 2 = 3 - 1
[3,1,2] => [2,3,1] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,3,2,4] => [1,2,4,3] => 3 = 4 - 1
[1,4,2,3] => [2,1,3,4] => 3 = 4 - 1
[2,1,3,4] => [1,3,2,4] => 3 = 4 - 1
[2,3,1,4] => [1,3,4,2] => 3 = 4 - 1
[2,4,1,3] => [3,1,2,4] => 3 = 4 - 1
[3,1,2,4] => [1,4,2,3] => 3 = 4 - 1
[3,4,1,2] => [4,1,2,3] => 3 = 4 - 1
[4,1,2,3] => [2,3,4,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[2,1,3,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[3,1,2,4,5] => [1,4,2,3,5] => 4 = 5 - 1
[4,1,2,3,5] => [1,5,2,3,4] => 4 = 5 - 1
[5,1,2,3,4] => [2,3,4,5,1] => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 6 = 7 - 1

Description

The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

Matching statistic: St000325
(load all 124 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 2 - 1
[1,2] => [2,1] => 2 = 3 - 1
[2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [3,2,1] => 3 = 4 - 1
[1,3,2] => [3,1,2] => 2 = 3 - 1
[2,1,3] => [2,3,1] => 2 = 3 - 1
[2,3,1] => [2,1,3] => 2 = 3 - 1
[3,1,2] => [1,3,2] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,3,2,4] => [4,2,3,1] => 3 = 4 - 1
[1,4,2,3] => [4,1,3,2] => 3 = 4 - 1
[2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[2,3,1,4] => [3,2,4,1] => 3 = 4 - 1
[2,4,1,3] => [3,1,4,2] => 3 = 4 - 1
[3,1,2,4] => [2,4,3,1] => 3 = 4 - 1
[3,4,1,2] => [2,1,4,3] => 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => 5 = 6 - 1
[2,1,3,4,5] => [4,5,3,2,1] => 4 = 5 - 1
[3,1,2,4,5] => [3,5,4,2,1] => 4 = 5 - 1
[4,1,2,3,5] => [2,5,4,3,1] => 4 = 5 - 1
[5,1,2,3,4] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => 6 = 7 - 1

Description

The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.

Matching statistic: St000470
(load all 124 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 2 - 1
[1,2] => [2,1] => 2 = 3 - 1
[2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [3,2,1] => 3 = 4 - 1
[1,3,2] => [3,1,2] => 2 = 3 - 1
[2,1,3] => [2,3,1] => 2 = 3 - 1
[2,3,1] => [2,1,3] => 2 = 3 - 1
[3,1,2] => [1,3,2] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,3,2,4] => [4,2,3,1] => 3 = 4 - 1
[1,4,2,3] => [4,1,3,2] => 3 = 4 - 1
[2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[2,3,1,4] => [3,2,4,1] => 3 = 4 - 1
[2,4,1,3] => [3,1,4,2] => 3 = 4 - 1
[3,1,2,4] => [2,4,3,1] => 3 = 4 - 1
[3,4,1,2] => [2,1,4,3] => 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => 5 = 6 - 1
[2,1,3,4,5] => [4,5,3,2,1] => 4 = 5 - 1
[3,1,2,4,5] => [3,5,4,2,1] => 4 = 5 - 1
[4,1,2,3,5] => [2,5,4,3,1] => 4 = 5 - 1
[5,1,2,3,4] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => 6 = 7 - 1

Description

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

Matching statistic: St000507
(load all 41 compositions to match this statistic)

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1]]
 => 1 = 2 - 1
[1,2] => [[1,2]]
 => 2 = 3 - 1
[2,1] => [[1],[2]]
 => 1 = 2 - 1
[1,2,3] => [[1,2,3]]
 => 3 = 4 - 1
[1,3,2] => [[1,2],[3]]
 => 2 = 3 - 1
[2,1,3] => [[1,3],[2]]
 => 2 = 3 - 1
[2,3,1] => [[1,2],[3]]
 => 2 = 3 - 1
[3,1,2] => [[1,3],[2]]
 => 2 = 3 - 1
[1,2,3,4] => [[1,2,3,4]]
 => 4 = 5 - 1
[1,3,2,4] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,4,2,3] => [[1,2,4],[3]]
 => 3 = 4 - 1
[2,1,3,4] => [[1,3,4],[2]]
 => 3 = 4 - 1
[2,3,1,4] => [[1,2,4],[3]]
 => 3 = 4 - 1
[2,4,1,3] => [[1,2],[3,4]]
 => 3 = 4 - 1
[3,1,2,4] => [[1,3,4],[2]]
 => 3 = 4 - 1
[3,4,1,2] => [[1,2],[3,4]]
 => 3 = 4 - 1
[4,1,2,3] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,2,3,4,5] => [[1,2,3,4,5]]
 => 5 = 6 - 1
[2,1,3,4,5] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[3,1,2,4,5] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[4,1,2,3,5] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[5,1,2,3,4] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
 => 6 = 7 - 1

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St001250
(load all 5 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => 1 = 2 - 1
[1,2] => [1,1]
 => 2 = 3 - 1
[2,1] => [2]
 => 1 = 2 - 1
[1,2,3] => [1,1,1]
 => 3 = 4 - 1
[1,3,2] => [2,1]
 => 2 = 3 - 1
[2,1,3] => [2,1]
 => 2 = 3 - 1
[2,3,1] => [2,1]
 => 2 = 3 - 1
[3,1,2] => [2,1]
 => 2 = 3 - 1
[1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[1,3,2,4] => [2,1,1]
 => 3 = 4 - 1
[1,4,2,3] => [2,1,1]
 => 3 = 4 - 1
[2,1,3,4] => [2,1,1]
 => 3 = 4 - 1
[2,3,1,4] => [2,1,1]
 => 3 = 4 - 1
[2,4,1,3] => [2,1,1]
 => 3 = 4 - 1
[3,1,2,4] => [2,1,1]
 => 3 = 4 - 1
[3,4,1,2] => [2,1,1]
 => 3 = 4 - 1
[4,1,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[2,1,3,4,5] => [2,1,1,1]
 => 4 = 5 - 1
[3,1,2,4,5] => [2,1,1,1]
 => 4 = 5 - 1
[4,1,2,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[5,1,2,3,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 6 = 7 - 1

Description

The number of parts of a partition that are not congruent 0 modulo 3.

The following 548 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001389The number of partitions of the same length below the given integer partition. St000021The number of descents of a permutation. St001176The size of a partition minus its first part. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000167The number of leaves of an ordered tree. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000443The number of long tunnels of a Dyck path. St000528The height of a poset. St000532The total number of rook placements on a Ferrers board. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000912The number of maximal antichains in a poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001286The annihilation number of a graph. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000148The number of odd parts of a partition. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000228The size of a partition. St000238The number of indices that are not small weak excedances. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001427The number of descents of a signed permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000026The position of the first return of a Dyck path. St000050The depth or height of a binary tree. St000393The number of strictly increasing runs in a binary word. St000439The position of the first down step of a Dyck path. St000626The minimal period of a binary word. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001313The number of Dyck paths above the lattice path given by a binary word. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001626The number of maximal proper sublattices of a lattice. St001674The number of vertices of the largest induced star graph in the graph. St001782The order of rowmotion on the set of order ideals of a poset. St001883The mutual visibility number of a graph. St000006The dinv of a Dyck path. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000153The number of adjacent cycles of a permutation. St000164The number of short pairs. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000374The number of exclusive right-to-left minima of a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000527The width of the poset. St000636The hull number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000746The number of pairs with odd minimum in a perfect matching. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000806The semiperimeter of the associated bargraph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000839The largest opener of a set partition. St000883The number of longest increasing subsequences of a permutation. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001963The tree-depth of a graph. St000005The bounce statistic of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000041The number of nestings of a perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000074The number of special entries. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000246The number of non-inversions of a permutation. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000691The number of changes of a binary word. St000731The number of double exceedences of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000992The alternating sum of the parts of an integer partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001358The largest degree of a regular subgraph of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001644The dimension of a graph. St001726The number of visible inversions of a permutation. St001777The number of weak descents in an integer composition. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001910The height of the middle non-run of a Dyck path. St001962The proper pathwidth of a graph. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000829The Ulam distance of a permutation to the identity permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St000144The pyramid weight of the Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000863The length of the first row of the shifted shape of a permutation. St000060The greater neighbor of the maximum. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000653The last descent of a permutation. St000729The minimal arc length of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001346The number of parking functions that give the same permutation. St001808The box weight or horizontal decoration of a Dyck path. St001820The size of the image of the pop stack sorting operator. St000039The number of crossings of a permutation. St000159The number of distinct parts of the integer partition. St000216The absolute length of a permutation. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000376The bounce deficit of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000710The number of big deficiencies of a permutation. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001152The number of pairs with even minimum in a perfect matching. St001684The reduced word complexity of a permutation. St001668The number of points of the poset minus the width of the poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000438The position of the last up step in a Dyck path. St000643The size of the largest orbit of antichains under Panyushev complementation. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000014The number of parking functions supported by a Dyck path. St000133The "bounce" of a permutation. St000294The number of distinct factors of a binary word. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001658The total number of rook placements on a Ferrers board. St001863The number of weak excedances of a signed permutation. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000335The difference of lower and upper interactions. St000531The leading coefficient of the rook polynomial of an integer partition. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000878The number of ones minus the number of zeros of a binary word. St000922The minimal number such that all substrings of this length are unique. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001613The binary logarithm of the size of the center of a lattice. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001884The number of borders of a binary word. St001896The number of right descents of a signed permutations. St001959The product of the heights of the peaks of a Dyck path. St000145The Dyson rank of a partition. St000295The length of the border of a binary word. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001955The number of natural descents for set-valued two row standard Young tableaux. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001637The number of (upper) dissectors of a poset. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000225Difference between largest and smallest parts in a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001875The number of simple modules with projective dimension at most 1. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000454The largest eigenvalue of a graph if it is integral. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St000744The length of the path to the largest entry in a standard Young tableau. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000815The number of semistandard Young tableaux of partition weight of given shape. St001570The minimal number of edges to add to make a graph Hamiltonian. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St000758The length of the longest staircase fitting into an integer composition. St000761The number of ascents in an integer composition. St001566The length of the longest arithmetic progression in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001928The number of non-overlapping descents in a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000260The radius of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000741The Colin de Verdière graph invariant. St000264The girth of a graph, which is not a tree. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000933The number of multipartitions of sizes given by an integer partition. St000997The even-odd crank of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001651The Frankl number of a lattice. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000736The last entry in the first row of a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001118The acyclic chromatic index of a graph. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation.